#244538

order	512
nilpotency class	3
isoclinism	isoclinic class of #244523
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#61778	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

977 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,7,12)(3,13,18,26)(4,19,22,29)(5,23,6,11)(8,17,27,16)(9,20,10,21)(14,30,31,24)(15,25,32,28)(33,34)(35,39)(36,40)(37,41)(38,42)(43,49)(44,50)(45,51)(46,59)(47,54)(48,55)(52,57)(53,60)(56,63)(58,62)(61,64);(1,3)(2,8)(4,15)(5,17)(6,16)(7,18)(9,28)(10,25)(11,26)(12,27)(13,23)(14,20)(19,24)(21,31)(22,32)(29,30)(33,35,38,45)(34,39,42,51)(36,43,47,57)(37,44,48,58)(40,52,54,49)(41,50,55,62)(46,56,60,64)(53,63,59,61);(1,4)(2,9)(3,14)(5,20)(6,21)(7,22)(8,24)(10,12)(11,29)(13,25)(15,16)(17,32)(18,31)(19,23)(26,28)(27,30)(33,36)(34,40)(35,43)(37,46)(38,47)(39,49)(41,53)(42,54)(44,56)(45,57)(48,60)(50,61)(51,52)(55,59)(58,64)(62,63);(1,4)(2,10)(3,15)(5,21)(6,20)(7,22)(8,25)(9,12)(11,29)(13,24)(14,16)(17,31)(18,32)(19,23)(26,30)(27,28)(33,37)(34,41)(35,44)(36,46)(38,48)(39,50)(40,53)(42,55)(43,56)(45,58)(47,60)(49,61)(51,62)(52,63)(54,59)(57,64)

Polycyclic presentation:

2^9|a^2=h,b^2=i,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,g^a=gi,c^b=cg,e^b=eh,g^b=gh,e^c=ei,f^c=fh,g^c=gh,e^d=eh,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151734);